Let us note that the Full-to-TT compression, the TTε recompression and the TT–rounding algorithms make heavy use of SVD and QR algorithms for matrices.which would seem to be a good candidate for insertion of robust PCA and randomized PCA work. There is also a toolbox to dwell into it:

TT-Toolbox (TT=Tensor Train) Ver­sion 2.2
TT(Tensor Train) for­mat is an effi­cient way for low-parametric rep­re­sen­ta­tion of high-dimensional ten­sors. The TT-Toolbox is a MATLAB imple­men­ta­tion of basic oper­a­tions with
ten­sors in TT-format. It includes: